Wikipedia:Reference desk/Archives/Mathematics/2019 February 28

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February 28

Rates of return

I recently encountered an article about some retirees and some MIT math majors who won millions of dollars on a Massachusetts Lottery game that was poorly designed; if you bought $1000 of tickets in the right circumstances, and the tickets were a perfect sample of the general "population" of tickets, you'd win more than $1000 in prizes. Obviously you could lose money on a specific batch of tickets, but per the law of large numbers, you were almost surely going to get the money back, since the lottery had accidentally designed the game so that the odds were against itself at a certain point. It's as if 1/1000 tickets win money, and the winning prize is 1500x the purchase price; each ticket has a tiny chance of being a winner, but if you buy enough tickets, you're almost surely going to profit by 50%. What do we call this percentage? Profit margin seems to be more specifically economic, as does Rate of return; I'm wondering if there's a mathematical term that would apply even if this weren't a matter of money. Can we say that 50% is the expected value, or have I misunderstood that article? I've never heard the term before, and the article's introduction has many links to terms I don't well understand. Nyttend (talk) 00:32, 28 February 2019 (UTC)

The expected value of the outcome of a random process is the average. For example, if a lottery has 96 tickets that win nothing, 3 that win $5 and 1 than wins $100, the expected value of a ticket is the average value ${\frac {0\cdot 96+3\cdot 5+1\cdot 100}{100}}=\$1.15$. In particular, for such a lottery, one could be guaranteed of making money if tickets cost less than $1.15 and one can purchase all of them. Note that in this toy example, even purchasing a single ticket is positive expected value, but almost certainly a losing proposition. The MIT situation is more complicated, involving spillovers from the jackpot onto lower-prize tickets. This meant that they didn't need to get very rare jackpot tickets to have positive expected value. --207.232.84.226 (talk) 12:09, 28 February 2019 (UTC)

• The terminology is that 50% is the expected value of the rate of return.Loraof (talk) 19:23, 28 February 2019 (UTC)

• Even with positive expectation the lottery is usually a bad deal unless you a have very large bankroll to start with, per the Kelly criterion. See: [1]. 173.228.123.166 (talk) 03:08, 2 March 2019 (UTC)
  • They did have a fairly large bankroll. The news article doesn't mention the price of the tickets, but implies it was $1. On their first try they bought $3,600 worth, enough to have an expectation of three 4-number winners and about sixty 3-number winners, according to the figures in the article. They actually won $6,300 and immediately started buying more tickets. (I mean, more tickets each time.) --Revised wording 76.69.46.228 (talk) 20:36, 3 March 2019 (UTC)